Although the spatial locality constraint comes at the cost of decreased performance in the rate
and relative distance, this scaling of the distance is nearly optimal. Several upper bounds have
been proven about the parameters of spatially local subsystem codes in \(D\)  dimensions. For
this case, Bravyi and Terhal [1]} have shown that \(d \le O(n^{1-1/D})\) . Our
codes nearly saturate this bound and have the virtue that they are in general constructive. In
particular, our codes in \(D=3\)  dimensions already improve on the previous best results (by
Ref. [2]} again) for arbitrary sparse codes and achieve \(d = n^{2/3-\epsilon }\) 
for \(\epsilon = O(1/\sqrt{\log {n}})\) .
